First, it is common practice to use Q (\(\displaystyle \mathbb{Q}\)) for the set of rational numbers.Let r = rational number
Let i = irrational number. Why does r + i = i?
First, it is common practice to use Q (\(\displaystyle \mathbb{Q}\)) for the set of rational numbers.
If \(\displaystyle x\in\mathbb{R}\) i.e. \(\displaystyle x\) is a real number & \(\displaystyle x\notin\mathbb{Q}\) the we say that \(\displaystyle x\) is irrational, [less common is \(\displaystyle x\in\mathscr{I}].\).
Say \(\displaystyle a\in\mathbb{Q}~\&~s\in\mathscr{I}\), one rational the other irrational.
If \(\displaystyle a+s\) were rational, lets say \(\displaystyle a+s=z\in\mathbb{Q}\) then \(\displaystyle s=z-a\).
However the difference of two rational numbers is rational. BUT \(\displaystyle s\) not rational.
No that is a false statement. In my division, the best professors were the ones who set out the first day of class what level of background was expected in the class. Samples are given out and students in the class are asked to sign that they understand the prerequisite material.Can you explain this at my level? This is what good teachers and tutors do.
No that is a false statement. In my division, the best professors were the ones who set out the first day of class what level of background was expected in the class. Samples are given out and students in the class are asked to sign that they understand the prerequisite material.
If you lack basic understanding of operation with rational numbers why are you asking questions about them?
Is it more important to win or more important to learn. Philosophy matters. Attitude matters.I cannot win with you, right? It was just a simple comment.
Here's an answer which is more of a visual demonstration than a rigorous explanation. But first, we need to remember the following patterns about decimal forms of rational and irrational numbers.Let r = rational number
Let i = irrational number
Why does r + i = i?
Is it more important to win or more important to learn. Philosophy matters. Attitude matters.
Here's an answer which is more of a visual demonstration than a rigorous explanation. But first, we need to remember the following patterns about decimal forms of rational and irrational numbers.
The decimal form of a rational number always ends with a repeating pattern of digits. Here are some examples:
12/5 = 2.4000... (repeats zeros forever)
4/9 = 0.4444... (repeats fours)
45/7 = 6.428571428571... (repeats group of digits)
All rational numbers show a repeating form, when written as a decimal number. On the other hand, irrational numbers written in decimal form never show a repeating pattern:
Pi = 3.141592653589... (digits continue without pattern)
e = 2.718281828459045... (no pattern)
Now, let's add a rational and an irrational: 12/5 + Pi
2.400000000000...
3.141592653589...
-------------------------------
5.541592653589...
Can you agree that adding a random string of digits to a repeating pattern of digits will never result in a repeating pattern? Such a sum will never represent a rational number.
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Is your textbook talking about closure properties for sets of numbers? (That's what pka was getting at.) I'm not sure what prompted the question.... exactly what I was looking for ...
Is your textbook talking about closure properties for sets of numbers? (That's what pka was getting at.) I'm not sure what prompted the question.
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"criticize" doesn't have to be pejorative. If you seek help from volunteer mathematics professionals, why not accept the help you receive - no matter how you choose to frame it at first glance? Take it at face value and learn from it. Simple.I am just tired of getting criticized for posting math questions, easy and not so easy, in a MATH SITE.
"criticize" doesn't have to be pejorative. If you seek help from volunteer mathematics professionals, why not accept the help you receive - no matter how you choose to frame it at first glance? Take it at face value and learn from it. Simple.
If the question comes from your algebra review, then my visual demonstration is probably not adequate. That is, there's a specific lesson to be learned about the properties of the set of rational numbers, if that's what motivates the question.
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Can you explain this at my level? This is what good teachers and tutors do.
No, you're ignoring me.Can we move on now?